Systems and methods for reducing the actuation voltage for electrostatic mems devices

ABSTRACT

Systems and methods to amplify the response of a MEMS micro-oscillator by driving the MEMS device at its electrical and mechanical resonance frequencies, simultaneously. This enhances the MEMS mechanical sensitivity to electrical excitation and increases the voltage across the MEMS capacitor. Moreover, using a combination of two input signals at different frequencies (beat signal) may be used to achieve double resonance in any MEMS device, even if its natural frequency is far from its electrical resonance.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit under 35 U.S.C. § 119(e) ofU.S. Provisional Application Ser. No. 62/531,803, filed Jul. 12, 2017,and titled “SYSTEMS AND METHODS FOR REDUCING THE ACTUATION VOLTAGE FORELECTROSTATIC MEMS DEVICES,” which is hereby incorporated by referencein its entirety.

BACKGROUND

Micro-electro-mechanical-systems (MEMS) have received great attentiondue to their great potential and unique characteristics. A MEMS (alsotermed MEMS device herein) is inexpensive for mass production, is small,fast, highly sensitive, has low noise-sensitivity, and requires a smallamount of power, especially when they are actuated electrostatically.This is the most common way to actuate a MEMS device. Even though thisactuation method consumes very low power, compared to other actuationmethods, an electrostatic MEMS device requires very high voltage to moveits structure. This need is one of the major challenges limiting theadoption of electrostatic MEMS in very promising applications such as RFswitches and MEMS resonator-based sensors. Therefore, there has been anincreasing interest in reducing the electrostatic MEMS input voltage.Reducing the air gap between the MEMS structure and the substrate,increasing the electrostatic actuation area, or reducing the MEMSstiffness, are just few methods that were proposed in the prior art toreduce the actuation voltage for electrostatic MEMS. However, most ofthese methods limit the operation of the MEMS and could be a source forother problems. For example, reducing the air gap or increasing the MEMSactuation area will increase the undesirable effect ofsqueeze-film-damping. Moreover, decreasing the MEMS stiffness reducesthe MEMS immunity to stiction.

SUMMARY

Embodiments herein provide novel methods to amplify the effect of theelectrostatic input voltage by triggering the MEMS electrical circuitresonance. The methods advantageously impose no restriction on the MEMSoperation, nor do the methods require a device configuration or designchange.

According to an embodiment, a method is provided for actuating anelectrostatic micro-electro-mechanical system (MEMS) micro-oscillatordevice, wherein the MEMS device has a natural mechanical resonancefrequency and an internal electrical resonance frequency. The methodtypically includes driving the MEMS device with a first alternatingcurrent (AC) signal, and simultaneously driving the MEMS device with asecond AC signal, wherein a frequency of the first AC signal is withinthe bandwidth of the internal electrical resonance frequency (within the3-db bandwidth of the internal electrical resonance frequency) andwherein a difference between the frequency of the first AC signal and afrequency of the second AC signal is near to or substantially the sameas the natural mechanical resonance frequency (e.g., within about 5% to10%). In certain aspects, the MEMS oscillator device includes anelectrode arrangement comprising a first electrode and a secondelectrode arranged parallel to the first electrode, wherein the secondelectrode is fixed and wherein at least a first end of the firstelectrode is able to move relative to the second electrode. In certainaspects, the natural mechanical resonance frequency ω_(n)=√(k/m), andthe internal electrical resonance frequency ω_(e)=1/√{square root over((L_(s)C_(o)))}, wherein L_(s) is the parasitic inductance of theelectrode arrangement, C_(o) is a nominal capacitance of the electrodearrangement, wherein m is an effective mass of the first electrode and kis a linear stiffness of the first electrode. In certain aspects, themethod further includes applying a direct current (DC) signal to thefirst electrode.

According to another embodiment, a method is provided for actuating anelectrostatic micro-electro-mechanical system (MEMS) micro-oscillatordevice, wherein the MEMS device has a natural mechanical resonancefrequency that is near to or substantially the same as an internalelectrical resonance frequency of the MEMS device. The method typicallyincludes driving the MEMS device with a first alternating current (AC)signal, wherein a frequency of the first AC signal is near to orsubstantially the same as the internal electrical resonance frequency orthe natural mechanical resonance frequency (e.g., within about 5% to10%).

Reference to the remaining portions of the specification, including thedrawings and claims, will realize other features and advantages of thepresent invention. Further features and advantages of the presentinvention, as well as the structure and operation of various embodimentsof the present invention, are described in detail below with respect tothe accompanying drawings. In the drawings, like reference numbersindicate identical or functionally similar elements.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

The detailed description is described with reference to the accompanyingfigures. The use of the same reference numbers in different instances inthe description and the figures may indicate similar or identical items.

FIG. 1 shows a typical configuration of a clamped-clamped parallel plateelectrostatic MEMS device.

FIG. 2 shows the change in capacitance of a MEMS device due to frequencyand parasitic inductance.

FIG. 3 shows a MEMS single degree of freedom model of a MEMS systemaccording to an embodiment.

FIG. 4A shows an experimental set-up of the MEMS device and FIG. 4Bshows the single degree of freedom model.

FIG. 5 shows the measured amplification (gain) for the voltage acrossthe MEMS with respect to the input voltage as a function of the ACfrequency, according to an embodiment.

FIG. 6 shows the experimental time history for the MEMS deflection forAC signals with and without double resonance amplification, according toembodiments.

FIG. 7A, FIG. 7B, FIG. 8A and FIG. 8B show examples of double resonanceexcitation experimental results.

DETAILED DESCRIPTION

The following detailed description is exemplary in nature and is notintended to limit the invention or the application and uses of theinvention. Furthermore, there is no intention to be bound by anyexpressed or implied theory presented in the following detaileddescription or the appended drawings.

Turning to the drawings, and as described in detail herein, embodimentsof the disclosure provide methods, devices and systems useful to actuatea MEMS device.

Parallel-plate electrostatic force is the common actuation method inMEMS. A typical configuration is represented in FIG. 1 for aclamped-clamped micro-beam. Two modes of operation can be utilized usingthis configuration—the capacitive and the dynamic forcing modes.

In the capacitive mode, an applied small alternating current (AC)voltage, V_(AC), is used to measure changes in the MEMS effectivecapacitance due to a condition of interest, such as water vapor(humidity). In the dynamic forcing mode, or dynamics mode, the movablestructure (upper electrode) is usually biased by an electrostatic directcurrent (DC) and AC load. The DC voltage, V_(DC), deflects the upper(movable) electrode slightly and V_(AC) vibrates the electrode aroundthe new, deflected position. Usually, the AC frequency is chosen suchthat it drives the MEMS microstructure at mechanical resonance, such asin resonant sensors. The large mechanical vibration of the MEMS atresonance provides higher sensitivity and better noise rejectioncompared to the static mode.

Capacitive mode

An ideal typical capacitive MEMS device includes two electricallyisolated parallel plates. The two parallel plates create an electricalcapacitor with a nominal capacitance C_(o) (F) given by:

$\begin{matrix}{C_{o} = \frac{ɛ_{o}ɛ_{r}A}{d}} & (1)\end{matrix}$

where A is the overlapping area of the parallel plates, ϵ_(o) is theelectric constant of the vacuum, and ϵ_(r) is the relative permittivityof the medium between the two plates. However, the internal connectionsin a physical capacitor always produces a parasitic inductor L_(s) (H),which is usually ignored at low excitation frequency. At high excitationfrequencies, below the MEMS circuit resonance frequency, the effectivecapacitance C_(E) of the MEMS circuit, described in equation (2), isalways higher than the nominal capacitance C_(o):

$\begin{matrix}{C_{E} = \frac{C_{o}}{\left( {1 - {\omega^{2}L_{s}C_{o}}} \right)}} & (2)\end{matrix}$

where ω is the excitation frequency of the AC voltage (rad/s).

Simulation for the capacitance amplification is shown in FIG. 2 for aMEMS device that will be introduced below. FIG. 2 shows that the MEMSeffective capacitance reaches very high values as the AC excitationfrequency is increased such that co approaches 1/√{square root over(L_(s)C_(o))}, the MEMS internal electrical resonance frequency. Thislarge amplification in the MEMS effective capacitance, and hence, itshigh sensitivity to changes in ϵ_(r), has been recently proposed todetect and measure tiny concentrations of analytes, such as measuringspecific protein content in a label-free manner. In certain embodiments,the capacitance amplification, near the MEMS electrical resonance, isused to amplify the effect of the AC input voltage during the dynamicsmode, as described herein.

Dynamics Mode

As stated above, the dynamics excitation of an electrostatic MEMS deviceis accomplished by suppling DC and AC voltages. However, a large voltageis usually required to vibrate the device. This high actuation voltagerequirement for electrostatic MEMS devices is the most significantdisadvantage of electrostatic actuation compared to other actuationmethods, such as piezo-electric and electromagnetics actuation. Evenwith this limitation, electrostatic actuation is the most commonactuation method used in MEMS applications. According to an embodiment,double resonance (mechanical and electrical resonance) is used toeliminate the need for high input voltage actuation.

In the following example, a single-degree-of-freedom spring-mass-dampersystem was adapted to model the dynamics of an electrostatic MEMSdevice, as shown in FIG. 3. The system is electrostatically actuated bya DC voltage signal superimposed with AC voltage. In this model, theMEMS mass is treated as a single point and it can only move in onedirection. The equation of motion of the system is given by:

m _(eff) {umlaut over (x)}(t)+c(x){dot over (x)}+kx=F _(e)(x, t)   (3)

where x is the MEMS deflection (m), the dot operators represent temporalderivatives, t is the time in seconds, m_(eff) (kg) is the effectivemass, given by m_(eff)=k/ω_(n) ², ω_(n) is the natural frequency of thesystem (rad/s), k is the linear stiffness of the micro beam (N/m), c(x)is the nonlinear squeeze film damping of the system (Ns/m) given byBlech model, and F_(e)(x, t) is the electrostatic force (N) that isusually given by:

$\begin{matrix}{{F_{e}\left( {x,t} \right)} = \frac{\epsilon_{o}\epsilon \; A\; \left( {V_{D\; C} + {V_{A\; C}{\cos \left( {\omega \; t} \right)}}} \right)^{2}}{2\left( {d - x} \right)^{2}}} & (4)\end{matrix}$

In equation 4, V_(DC) and V_(AC) are the input DC voltage and input ACvoltage amplitude, respectively. Under common actuation conditions, theinput AC frequency is much lower than the MEMS electrical resonancefrequency (ω_(e)=1/√{square root over (L _(s) C _(o)))}, due to the lowparasitic inductor value. Thus, the voltage across the MEMS is assumedequal to the input voltage, as shown in FIG. 3. While MEMS devices arecommonly connected to resistors in series as a precaution to pull-in,under high input AC excitation frequencies, a series RLC(Resistor-inductor-capacitor) circuit analysis is required to calculatethe voltage across the MEMS capacitor, V_(ACeff)(x). The amplitude ofthis voltage is calculated by:

$\begin{matrix}{{V_{ACeff}(x)} = \frac{V_{A\; C}}{\left( \sqrt{\left( {\omega \; {{RC}(x)}} \right)^{2} + \left( {1 - {\omega^{2}*L*{C(x)}}} \right)^{2}} \right)}} & (5)\end{matrix}$

where R is the MEMS circuit resistance value, L is the total seriesinductance of the circuit and equals L_(external)+L_(s), and C(x) is theMEMS capacitance value which is a function of x given by:

$\begin{matrix}{{C(x)} = \frac{ɛ\; A}{\left( {d - x} \right)}} & (6)\end{matrix}$

Substituting equation (5) into equation (4) then back into equation (3)gives:

$\begin{matrix}{{{m\; \overset{¨}{x}} + {{c(x)}\overset{.}{x}} + {kx}} = {\frac{ɛ\; A}{2\left( {d - x} \right)^{2}}\left\lbrack {V_{D\; C} + \frac{V_{A\; C}{\cos \left( {\omega \; t} \right)}}{\sqrt{\left. {\left( {\omega \; {{RC}(x)}} \right)^{2} + \left( {1 - {\omega^{2}{{LC}(x)}}} \right)^{2}} \right)}}} \right\rbrack}^{2}} & (7)\end{matrix}$

Equation (7) is a highly coupled equation; the voltage amplification(due to circuit resonance) is a function of the MEMS deflection, x (dueto mechanical resonance). The MEMS deflection, x, is a function of theeffective voltage. Moreover, for a MEMS circuit with a small R-value(circuit damping), the effective voltage can be amplified by over anorder of magnitude of the input voltage, if the frequency is chosen tobe near the MEMS electrical circuit resonance. This large voltageamplification can relax the need for the high input voltage to vibratean electrostatic MEMS device if the MEMS natural mechanical resonancefrequency, ω_(n)=√(k/m)is also near ω_(e). This interesting phenomenonis termed herein as double resonance actuation. This phenomenon can bealternatively achieved for a MEMS device that has a large differencebetween ω_(e) and ω_(n) (as is the case for the MEMS device discussed inthe following section) by exciting the MEMS device with two different ACsources of frequency ω₁ and ω₂. To produce the double resonanceamplification, at least one of the excitation signal frequencies shouldbe near ω_(e), and the absolute difference Δω=∥ω₁−ω₂∥ should be nearω_(n).

This special arrangement for double resonance activation is explained asfollows. First, assume the AC₁ signal has a frequency of ω₁ that is nearω_(e) and the difference between the two AC signal frequencies,Δω=∥ω₁−ω₂∥ is close to the mechanical resonance of the MEMS ω_(n). TheDC term (i.e. there is no applied DC voltage) is dropped to simplify theanalysis. Using the circuit superposition principle, equation (5) can beused to calculate the effective voltage for each signal separately. TheAC signal with frequency ω₁ will produce a voltage amplification givenby equation (5).

The second voltage signal may or may not produce voltage amplification.For a typical MEMS device, the mechanical natural frequency ω_(n) ishigh, such that Δω used is enough to push the frequency of the second ACsignal out of the electrical amplification bandwidth, thusV_(AC2eff)≅V_(AC2). The combined effect of the AC signals on the MEMSdynamics response can be described as:

$\begin{matrix}{{{m\; \overset{¨}{x}} + {{c(x)}\overset{.}{x}} + {kx}} = \frac{ɛ\; {A\left( {{V_{A\; C\; 2}{\cos \left( {\omega_{2}t} \right)}} + {{V_{A\; {Ceff}\; 1}(x)}\left( {\omega_{1}t} \right)}} \right)}^{2}}{2\left( {d - x} \right)^{2}}} & (8)\end{matrix}$

Expanding the voltage square term in (8) results in:

(V _(AC2)cos(ω₂ t)+V _(ACeff1)(x)(ω₁ t))²=

V _(ACeff1)cos²(ω₁ t)+V _(AC2) ²cos²(ω₂ t)

+V _(ACeff1) V _(AC2)(cos(ω₁+ω₂)+cos(ω₁−ω₂))   (9)

The term V_(ACeff1)V_(AC2)cos(ω₁−ω₂) in equation (9) is the only termthat will cause significant mechanical vibration; the difference ω₁−ω₂is near the mechanical vibration of the MEMS and V_(ACeff1) is anamplified version of V_(AC1), due to the MEMS internal circuitresonance. Dropping the other terms, equation (9) can be simplified to:

$\begin{matrix}{{{m\; \overset{¨}{x}} + {{c(x)}\overset{.}{x}} + {kx}} = \frac{ɛ\; {AV}_{A\; {Ceff}\; 1}V_{A\; C\; 2}{\cos \left( {\omega_{1} - \omega_{2}} \right)}}{2\left( {d - x} \right)^{2}}} & (10)\end{matrix}$

Equation (10) highlights that the effect of an AC input to a MEMS devicecan be amplified by over an order of magnitude by driving the MEMSdevice with two AC signals of frequency difference equal to the MEMSmechanical natural frequency and the frequency of at least one of thesesignals is near the MEMS internal circuit resonance frequency.

In an embodiment, a method for actuating an electrostaticmicro-electro-mechanical system (MEMS) micro-oscillator device includes,in a first step, providing the MEMS device. The MEMS oscillator devicemay include an electrode arrangement comprising a first electrode and asecond electrode arranged parallel to the first electrode, wherein thesecond electrode is fixed and wherein at least a first end of the firstelectrode is able to move relative to the second electrode. The MEMSdevice may be connected to one or more AC sources and a DC source. In asecond step, the MEMS device is driven with a first alternating current(AC) signal (e.g., from a first AC source); and simultaneously drivenwith a second AC signal (e.g., from a second AC source). The frequencyof the first AC signal is within the 3-db bandwidth of, or substantiallythe same as, the internal electrical resonance frequency of the MEMSdevice and a difference between the frequency of the first AC signal anda frequency of the second AC signal is near to or substantially the sameas the natural mechanical resonance frequency of the MEMS device. Thenatural mechanical resonance frequency ω_(n)=√(k/m), and the internalelectrical resonance frequency ω_(e)=1/√{square root over(L_(s)C_(o)))}, wherein L_(s) is the parasitic inductance of theelectrode arrangement, wherein C_(o) is a nominal capacitance of theelectrode arrangement, wherein m is an effective mass of the firstelectrode and wherein k is a linear stiffness of the first electrode. Inanother step, a direct current (DC) signal is also applied to the firstelectrode.

In another embodiment, a method for actuating an electrostaticmicro-electro-mechanical system (MEMS) micro-oscillator device includes,in a first step, providing the MEMS device. The MEMS oscillator devicemay include an electrode arrangement comprising a first electrode and asecond electrode arranged parallel to the first electrode, wherein thesecond electrode is fixed and wherein at least a first end of the firstelectrode is able to move relative to the second electrode. The MEMSdevice has a natural mechanical resonance frequency that is near to orsubstantially the same as an internal electrical resonance frequency ofthe MEMS device. The MEMS device may be connected to an AC source and aDC source. In a second step, the MEMS device is driven with a firstalternating current (AC) signal, wherein a frequency of the first ACsignal is near to or substantially the same as the internal electricalresonance frequency or the natural mechanical resonance frequency. Inanother step, a direct current (DC) signal is also applied to the firstelectrode.

Experimental validation

A Sensata™ MEMS accelerometer sensor has been used to validate thevoltage amplification. The device is shown in FIG. 4A and is describedin detail in Alsaleem, F. M., Younis, M. I., & Ouakad, H. M. (2009), Onthe nonlinear resonances and dynamic pull-in of electrostaticallyactuated resonators, Journal of Micromechanics and Microengineering. Themechanical resonance of the MEMS device was determined to be around 195Hz using a frequency sweep-method. The device has dimensions of l=9000μm and b=5320 μm. Due to the size of the MEMS sensor used, it is notpossible to achieve an oscillatory response at high pressure. Therefore,the experiment was carried out at reduced pressure of 8 Pa using avacuum chamber. The MEMS was driven using a Lyncee Tec stroboscopicmodule and its deflection was measured using Lyncee Tec's DHM (DigitalHolographic Microscope). The DHM measures the out-of-plane motion usingthe principle of light interference. Knowing the frequency of the MEMSoutput, the DHM synchronizes rapid, narrow laser pulses at the outputfrequency of the microbeam to take still images of the microbeam at afixed location. This allows the DHM's camera to take a nearly stillpicture of the microbeam at each frame. After n frames, the DHM measuresa complete cycle of oscillation. These post-processing steps areperformed using a stroboscopic module installed in a mainframe.Moreover, the MEMS capacitor is connected to a LabVIEW data acquisitioncard to monitor its voltage.

It was found experimentally that the internal circuit resonance of theMEMS is far from its mechanical resonance and from the MEMS analyzersampling frequency capability (1 Hz-25 MHz). To tune the circuitresonance of the MEMS, an external inductor L_(external)=-27 mH wasadded in parallel to the MEMS device. The added inductor shifts the MEMSelectrical resonance to be around 63.4 kHz. A resistor of 470 Ω wasadded in series to prevent the flow of large current in the circuitwhile still exciting the MEMS near the circuit resonance frequency.

Using the experimental set-up described above, an experiment wasconducted to confirm the voltage amplification across the MEMS devicedue to an RLC circuit resonance effect. In this experiment, a smallsignal with a frequency near the MEMS circuit resonance frequency wassupplied. Next, the frequency of the input signal was slowly swept whilemeasuring the voltage across the MEMS using the Lab VIEW acquisitioncard. FIG. 5 shows the measured amplification (gain) for the voltageacross the MEMS with respect to the input voltage. It is clear from FIG.5 that a voltage amplification of up to 21 times can be achieved bysupplying an AC signal with a frequency near the MEMS circuit resonance.If desirable, a higher amplification can be achieved using a smallerresistor value.

Next, an experiment was conducted to quantify the effect of doubleresonance voltage amplification on the MEMS mechanical vibration. Inthis experiment, the MEMS device was supplied with a single AC signal offrequency=190 Hz (near the MEMS natural mechanical resonance frequency).The signal has an amplitude of 2V. Next, the MEMS was supplied with twoAC signals, each with amplitude of only 1V. The first signal has afrequency of 60.8 kHz (near the electrical resonance frequency) and thedifference between the two signals is 190 Hz (near the mechanicalresonance frequency). From FIG. 5, a voltage amplification gain of up to8 times is expected, due to the double resonance effect. FIG. 6 comparesthe actual MEMS deflection measured by the DHM MEMS analyzer for theabove two cases. The figure clearly shows a large amplification in theMEMS response due to the double resonance effect; even though the totalinput AC voltage amplitude is equal in both cases.

FIG. 7A, FIG. 7B, FIG. 8A and FIG. 8B show examples of double resonanceexcitation experimental results.

FIG. 7A shows the frequency response of the system when excitedclassically by a single AC source at a frequency range of (85 Hz-107.5Hz) at 2 V. The maximum amplitude is about 0.1 μm. Experimental resultsare represented by circles and simulation is represented by a solidline. Also shown is the double resonance excitation using a beatingsignal composed of two voltage sources each with an amplitude of 1 V anda frequency of f₁=60.8 kHz and f₂=f₁−Δf. Experimental results are shownwith crosses and simulation is shown by a dashed line. The maximumamplitude at 195 Hz is equal to 1.3 μm. An amplification of 13 times isobserved in this case. Also, shown in FIG. 7B are the time histories ofthe vibration at 180 Hz, 190 Hz and 195 Hz, experimentally (dashed line)and theoretically (solid line).

FIG. 8A shows experimental and numerical simulation frequency responsewhen excited classically by a single AC source at a frequency range of(85 Hz-107.5 Hz) at 2 V. The maximum amplitude is about 0.1 μm.Experimental results are represented by circles and simulation isrepresented by a solid line. Also, shown is the double resonanceexcitation using a beating signal composed of two voltage sources botheach with an amplitude of 1 V and a frequency of f₁=63.1 kHz andf₂=f₁+Δf. Experimental results are shown with crosses and simulation isshown by a dashed line. The maximum amplitude equals 3.16 μm at 195 Hz,where an amplification of 30 times is observed in this case. Also, shownin FIG. 8B are the time histories of the vibration at 180 Hz, 190 Hz and195 Hz, experimentally (dashed line) and theoretically (solid line).

All references, including publications, patent applications, andpatents, cited herein are hereby incorporated by reference to the sameextent as if each reference were individually and specifically indicatedto be incorporated by reference and were set forth in its entiretyherein.

The use of the terms “a” and “an” and “the” and “at least one” andsimilar referents in the context of describing the disclosed subjectmatter (especially in the context of the following claims) are to beconstrued to cover both the singular and the plural, unless otherwiseindicated herein or clearly contradicted by context. The use of the term“at least one” followed by a list of one or more items (for example, “atleast one of A and B”) is to be construed to mean one item selected fromthe listed items (A or B) or any combination of two or more of thelisted items (A and B), unless otherwise indicated herein or clearlycontradicted by context. The terms “comprising,” “having,” “including,”and “containing” are to be construed as open-ended terms (i.e., meaning“including, but not limited to,”) unless otherwise noted. Recitation ofranges of values herein are merely intended to serve as a shorthandmethod of referring individually to each separate value falling withinthe range, unless otherwise indicated herein, and each separate value isincorporated into the specification as if it were individually recitedherein. All methods described herein can be performed in any suitableorder unless otherwise indicated herein or otherwise clearlycontradicted by context. The use of any and all examples, or examplelanguage (e.g., “such as”) provided herein, is intended merely to betterilluminate the disclosed subject matter and does not pose a limitationon the scope of the invention unless otherwise claimed. No language inthe specification should be construed as indicating any non-claimedelement as essential to the practice of the invention.

Certain embodiments are described herein. Variations of thoseembodiments may become apparent to those of ordinary skill in the artupon reading the foregoing description. The inventors expect skilledartisans to employ such variations as appropriate, and the inventorsintend for the embodiments to be practiced otherwise than asspecifically described herein. Accordingly, this disclosure includes allmodifications and equivalents of the subject matter recited in theclaims appended hereto as permitted by applicable law. Moreover, anycombination of the above-described elements in all possible variationsthereof is encompassed by the disclosure unless otherwise indicatedherein or otherwise clearly contradicted by context.

1. A method of actuating an electrostatic micro-electro-mechanicalsystem (MEMS) micro-oscillator device, wherein the MEMS device has anatural mechanical resonance frequency and an internal electricalresonance frequency, the method comprising: driving the MEMS device witha first alternating current (AC) signal; and simultaneously driving theMEMS device with a second AC signal, wherein a frequency of the first ACsignal is within the 3-db bandwidth of, or substantially the same as,the internal electrical resonance frequency and wherein a differencebetween the frequency of the first AC signal and a frequency of thesecond AC signal is near to or substantially the same as the naturalmechanical resonance frequency.
 2. The method of claim 1, wherein theMEMS oscillator device includes an electrode arrangement comprising afirst electrode and a second electrode arranged parallel to the firstelectrode, wherein the second electrode is fixed and wherein at least afirst end of the first electrode is able to move relative to the secondelectrode.
 3. The method of claim 2, wherein the natural mechanicalresonance frequency ω_(n)=√(k/m), and wherein the internal electricalresonance frequency ω_(e)=1/√{square root over (L_(s)C_(o)))}, whereinLs is the parasitic inductance of the electrode arrangement, whereinC_(o) is a nominal capacitance of the electrode arrangement, wherein mis an effective mass of the first electrode and wherein k is a linearstiffness of the first electrode.
 4. The method of claim 2, furthercomprising applying a direct current (DC) signal to the first electrode.5. A method of actuating an electrostatic micro-electro-mechanicalsystem (MEMS) micro-oscillator device, wherein the MEMS device has anatural mechanical resonance frequency that is near to or substantiallythe same as an internal electrical resonance frequency of the MEMSdevice, the method comprising: driving the MEMS device with a firstalternating current (AC) signal, wherein a frequency of the first ACsignal is near to or substantially the same as the internal electricalresonance frequency or the natural mechanical resonance frequency. 6.The method of claim 5, wherein the MEMS oscillator device includes anelectrode arrangement comprising a first electrode and a secondelectrode arranged parallel to the first electrode, wherein the secondelectrode is fixed and wherein at least a first end of the firstelectrode is able to move relative to the second electrode.
 7. Themethod of claim 6, further comprising applying a direct current (DC)signal to the first electrode
 8. A method of actuating an electrostaticmicro-electro-mechanical system (MEMS) micro-oscillator device by doubleresonance actuation as substantially described herein.